What is ln(5)+1/5 ln(4-x^2)-ln(x) ?

The solution to ln(5)+1/5 ln(4-x^2)-ln(x) is ln((5(4-x^2)^{1/5})/x)

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simplify ln(5)+1/5 ln(4-x^2)-ln(x)

simplify

ln (5) + \frac{1}{5} ln (4 - x^{2}) - ln (x)

ln \frac{5 ( 4 - x ^{2} ) ^{\frac{1}{5}}}{x}

Solution steps

ln (5) + \frac{1}{5} ln (4 - x^{2}) - ln (x)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{5} ln (- x^{2} + 4) = ln ((- x^{2} + 4)^{\frac{1}{5}})

= ln (5) + ln ((- x^{2} + 4)^{\frac{1}{5}}) - ln (x)

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (5) + ln ((4 - x^{2})^{\frac{1}{5}}) = ln (5 (- x^{2} + 4)^{\frac{1}{5}})

= ln (5 (4 - x^{2})^{\frac{1}{5}}) - ln (x)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (5 (4 - x^{2})^{\frac{1}{5}}) - ln (x) = ln \frac{5 ( - x ^{2} + 4 ) ^{\frac{1}{5}}}{x}

= ln \frac{5 ( 4 - x ^{2} ) ^{\frac{1}{5}}}{x}

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What is ln(5)+1/5 ln(4-x^2)-ln(x) ?
The solution to ln(5)+1/5 ln(4-x^2)-ln(x) is ln((5(4-x^2)^{1/5})/x)